↑

Properties

Label 2-21-21.20-c5-0-1
Degree $2$
Conductor $21$
Sign $-0.990 + 0.134i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.57i·2-s + (−13.5 + 7.74i)3-s + 0.942·4-s − 46.2·5-s + (−43.1 − 75.3i)6-s + (−120. − 48.7i)7-s + 183. i·8-s + (123. − 209. i)9-s − 257. i·10-s + 285. i·11-s + (−12.7 + 7.29i)12-s + 1.14e3i·13-s + (271. − 669. i)14-s + (625. − 357. i)15-s − 992.·16-s + 967.·17-s + ⋯
L(s)  = 1  + 0.985i·2-s + (−0.867 + 0.496i)3-s + 0.0294·4-s − 0.826·5-s + (−0.489 − 0.854i)6-s + (−0.926 − 0.375i)7-s + 1.01i·8-s + (0.506 − 0.862i)9-s − 0.814i·10-s + 0.711i·11-s + (−0.0255 + 0.0146i)12-s + 1.87i·13-s + (0.370 − 0.913i)14-s + (0.717 − 0.410i)15-s − 0.969·16-s + 0.812·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.990 + 0.134i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0467622 - 0.692566i\)
\(L(\frac12)\) \(\approx\) \(0.0467622 - 0.692566i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 - 7.74i)T \)
7 \( 1 + (120. + 48.7i)T \)
good2 \( 1 - 5.57iT - 32T^{2} \)
5 \( 1 + 46.2T + 3.12e3T^{2} \)
11 \( 1 - 285. iT - 1.61e5T^{2} \)
13 \( 1 - 1.14e3iT - 3.71e5T^{2} \)
17 \( 1 - 967.T + 1.41e6T^{2} \)
19 \( 1 + 112. iT - 2.47e6T^{2} \)
23 \( 1 + 1.26e3iT - 6.43e6T^{2} \)
29 \( 1 - 598. iT - 2.05e7T^{2} \)
31 \( 1 + 2.46e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.15e3T + 6.93e7T^{2} \)
41 \( 1 - 1.64e4T + 1.15e8T^{2} \)
43 \( 1 - 5.10e3T + 1.47e8T^{2} \)
47 \( 1 + 1.77e4T + 2.29e8T^{2} \)
53 \( 1 - 3.00e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.99e4T + 7.14e8T^{2} \)
61 \( 1 + 3.69e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.46e4T + 1.35e9T^{2} \)
71 \( 1 - 4.36e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.17e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.97e4T + 3.07e9T^{2} \)
83 \( 1 + 2.29e4T + 3.93e9T^{2} \)
89 \( 1 - 6.36e4T + 5.58e9T^{2} \)
97 \( 1 - 1.12e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.14251693223165423959898450882, −16.36643877472991092244325070876, −15.67917826026130500900862646630, −14.40036705373563786417938658646, −12.30127219416234376707656850104, −11.20497919835447054787048906059, −9.507905412429283519125579568382, −7.36263101867796782316713956470, −6.28395892457475593604487479267, −4.31905631515313288723733751327, 0.55632725601983360977821292890, 3.23636324193975020888997313609, 5.92634868520121605833532746188, 7.69316087628549682931422555743, 10.09516282042823854241687804937, 11.23799453891524722916189553414, 12.31175355833849203268264358395, 13.07344510448401444538989211717, 15.54534087502690816512824162130, 16.35620488139053058612780515009

Graph of the $Z$-function along the critical line